In this thesis we consider Itô-Lévy processes as driving processes. The derivation of such processes is highly complex, however, we will try to give the reader a feeling about their construction. We will also present conditions that ensure existence of such processes. Our approach to introduce Itô-Lévy processes will be to give a presentation of general Lévy processes, and then give a short but fairly thorough introduction of Lévy integrals. Then we state the definition of Itô-Lévy processes, and we will see that Lévy integrals are special cases of such processes.
General Lévy processes
We start this section by introducing maybe the most important result in the theory of Lévy processes. That is the Itô-Lévy decomposition, which gives a way of representing general Lévy processes.
Theorem 2.3.1 (Itô-Lévy decomposition, [ØS07]).LetL(t)be a Lévy process.
ThenL(t)has the decomposition
L(t) =αt+σW(t) + Z
R
xN¯(t, dx),
for some constantsα, σ∈R. W(t)is a Brownian motion such thatW(t)⊥ N˜(t, U), where the compensated Poisson random measureN(t, U˜ )is a mar-tingale as long asN(t, U)is defined as in Def. 2.2.3.
Finiteness of moments and exponential moments is very important when it comes to applications of Lévy processes. That is, it is important to be sure that characteristics such as mean, variance and moment generating functions exist. Fortunately, there are simple conditions that ensure these properties for Lévy processes.
Theorem 2.3.2 (Finiteness of moments end exponential moments of Lévy processes, [Ebe14]).Letp∈R. Then a Lévy processL(t)has
• finite absolutep-th moment if and only if Z
|x|≥R
|x|pν(dx)<∞;
2.3. General Lévy processes and Itô-Lévy processes
• finite exponentialp-th moment if and only if Z
|x|≥R
epxν(dx)<∞.
It is also worth mentioning the condition for finite variation of almost all paths of Lévy processes, because it is a necessary condition in the derivations of Ch. 4.
Theorem 2.3.3 (Finite variation of a.a. paths of Lévy processes, [Ebe14]).
A Lévy processL(t)has finite variation for a.a. paths ifσ= 0and Z
|x|<R
|x|ν(dx)<∞. (2.2)
A.a. paths ofL(t)have infinite variation ifσ6= 0or if Eq. 2.2 does not hold.
Remark ([Ebe14]).If the condition in Eq. 2.2 is satisfied, the small jumps in the Itô-Lévy decomposition converges, and we can split the small jump integral as
Z t 0
Z
R
x1{|x|<R}N˜(dt, dx) = Z t
0
Z
R
x1{|x|<R}N(dt, dx)−t Z
R
x1{|x|<R}ν(dx).
As mentioned earlier we are dependent on that processes modeling the underlying discounted financial assets prices are martingales in no-arbitrage frameworks. The next theorem gives us an opportunity to exploit the martingale property ofN˜(t, U). That is, the following theorem gives a way of considering Lévy processes where the jump part only consists of N(t, U˜ ), and thus it is straight forward to pinpoint when the Lévy process is a martingale.
Theorem 2.3.4 (Itô-Lévy decomposition whenR=∞, [ØS07]).IfEh L(1)
i<
∞we have that
Z
|x|≥R
|x|ν(dx)<∞, and we may then chooseR=∞such that
L(t) =αt+σW(t) + Z
R
xN˜(t, dx) forα,σ,W(t)andN(t, x)˜ as in Thm. 2.3.1.
Remark.Ifα= 0, thenL(t)is called a Lévy martingale.
Another essential theorem in the theory of Lévy processes is the Lévy-Khintchine formula, which gives us an easy way to find the characteristic function of any given Lévy process. We will use Lévy-Khintchine formula in the proof of Thm. 2.4.3, which states a formula for the characteristic function of exponential Lévy integrals.
Theorem 2.3.5 (Lévy-Khintchine formula, [ØS07], [Ebe14]).Let L(t) be a Lévy process with Lévy measureν. Then forθ∈R
Eh eiθL(t)i
=etψ(θ), where
ψ(θ) =iαθ−1 2σ2θ2+
Z
R
eiθx−1−1{|x|<R}iθx
ν(dx) (2.3) is called the Lévy symbol ofL(t)whenR
Rmin(1, x2)ν(dx)<∞. Conversely, given constantsα,σ2and a measureν such thatR
Rmin(1, x2)ν(dx)<∞, Eq.
(2.3) is the Lévy symbol of some Lévy processL(t). (α, σ2, ν)is called the triplet characterizing the Lévy process.
Examples of Lévy processes
We have already introduced the Brownian motion, which is an important special case of Lévy processes, and now we are going to introduce further two important examples.
Example 2.3.1 (Poisson processes, [App04],[ØS07]).A Lévy processL(t) = N(t)is called a Poisson process of intensityλ >0if it takes values inN∪ {0}
such that
P N(t) =n
=(λt)n n! e−λt.
That is,N(t)is a Poisson random variable with meanλt. Notice that the Poisson process is such that
N(t) =N(t, U = 1, ω),
which means that N(t)is a Poisson random measure with intensityλ= ν(U = 1).
In Ch. 5 we consider a specific stochastic volatility process, and in Ch. 6 we will analyze this stochastic volatility explicitly. The Lévy process in the following example (when it has exponential jump sizes) will be the driver of the stochastic volatility in that explicit case.
Example 2.3.2 (Compound Poisson processes (CPP), [ØS07]).LetY(n), for n∈N, be a sequence of i.i.d. random variables with values inRand common lawµY. Also, letN(t)be a Poisson process with intensityλ, such that it is independent of eachY(n). A Lévy processL(t) =Z(t)is called a CPP if it has the form
Z(t) =
N(t)
X
n=1
Y(n)
for eacht≥0. An increment of the process can be expressed as Z(s)−Z(t) =
N(s)
X
n=N(t)+1
Y(n)
2.3. General Lévy processes and Itô-Lévy processes fors > t, and its Lévy measureν is given by
ν(U) =E
N(1, U)
=λµY.
Remark.Notice thatZ(t) =N(t)ifY(n) := 1∀n.
Subordinators
In Ch. 5 we introduce a stochastic volatility driven by what is called a subordinator. We define such processes here, as well as a theorem giving their characteristic function.
Definition 2.3.1 (Subordinator, [BBK08]).A monotonically increasing Lévy processes is called a subordinator.
The following corollary, which states the formula for the characteristic function of subordinators, is the the Lévy-Khintchine formula with a specific Lévy symbol. This specific Lévy symbol is needed when we want to find the characteristic function of a subordinator which is defined in Ch. 5.
Corollary 2.3.1 (Characteristic function of subordinators, [App04]).Let L(t)be a subordinator with Lévy measureν. Then the Lévy symbol takes the form
ψ(θ) =iαθ+ Z ∞
0
eiθx−1
ν(dx), (2.4)
whereθ∈R,α≥0,ν(−∞,0] = 0andR
Rmin(1, x2)ν(dx)<∞. Conversely, given a constant α ≥ 0 and a measure ν such that ν(−∞,0] = 0 and R
Rmin(1, x2)ν(dx) < ∞, Eq. (2.4) is the Lévy symbol of some subordina-torL(t).
Integrals with respect to Lévy processes
A short introduction of Itô integrals was presented in the preceding. The integral with respect to general Lèvy processes is also an important tool in stochastics, and will be introduced next. As mentioned earlier Lévy integrals are special cases of Itô-Lévy processes, and we will use this special case to model instantaneous forward rates in Ch. 4. We will keep the intro-duction to the Lévy integrals short, but thorough enough for the reader to understand what kind of processes that are Lévy integrable.
First we define the mode of convergence required on the space of Lévy integrable processes.
Definition 2.3.2 (Processes that uniformly converges on compacts in prob-ability, [Low09]).A sequence of jointly measurable stochastic processes Xn(t)are said to uniformly converge on compacts in probability (ucp) to the limitX(t)if
P sup
s≤t
Xn(s)−X(s) >
!
→0
asn→ ∞,∀tand >0.
Next we define two function spaces. In Thm. 2.3.7 we will see that Lévy integrable processes are contained in one of these spaces, and that the Lévy integrals are contained in the second.
Definition 2.3.3 (The function spacesLucpandDucp, [ØS07]).
• DefineLucpas the space of cáglád adapted processes which are ucp.
• DefineDucpas the space of cádlág adapted processes which are ucp.
By Assum. 2.2.1 all Lévy processes in this thesis are their càdlàg version. The next theorem then ensures that the considered Lévy processes are semimartingales as well.
Theorem 2.3.6 (Lévy process as a semimartingale, [Low10]).Every càdlàg Lévy process is a semimartingale.
Now we have what we need to state the theorem which gives the neces-sary conditions to define an integral with respect to general Lévy processes.
Theorem 2.3.7 (Integral with respect to general Lévy processes, [ØS07]).
If the stochastic processH(t, T)is such thatH(t, T)∈ Lucp and the Lévy processL(t)is a semimartingale we can define the stochastic integral
X(t) = Z t
0
H(s, T)dL(s), (2.5)
whereX(t)is a continuous linear map
X(t) :Lucp→Ducp. Itô-Lévy processes
Following [ØS07], we consider Thm. 2.3.7, and observe that we can split the integral in Eq. (2.5) into three terms. That is, we can split it into three terms which are integrals with respect todt,dW(t)andN¯(dt, dx). By this observation, and by having the Itô-Lévy decomposition (Thm. 2.3.1) in mind, it is natural to think that more general SDE’s of the form
dL(t) =ˆ α(t, T, ω)dt+σ(t, T, ω)dW(t) + Z
R
γ(t, T, x, ω)N¯(dt, dx) (2.6) are possible to define. As long as the coefficient processesα(t, T), σ(t, T) andγ(t, T, x)satisfy certain conditions such that the integrals exist, they are indeed possible to define. Processes of the form as in Eq. (2.6) are called Itô-Lévy processes, and they are used as driving processes in this thesis.
We note that general Lévy processes and Lévy integrals are special cases of Itô-Lévy processes.
Notation 2.3.1.Itô-Lévy processes are denoted byL(t).ˆ
2.3. General Lévy processes and Itô-Lévy processes Because we use Itô-Lévy processes as driving processes in this thesis, the next definition will be important to ensure that those processes are well defined.
Definition 2.3.4 (The function spaceU).Given an Itô-Lévy processL(t), letˆ U∗=U∗([0,T]3×[0,T]3×U)be the class of triplets(α(t, T), σ(t, T), γ(t, T, x)) such that
1. α(t, T, ω), σ(t, T, ω) : [0,T]2 ×Ω → R are B2 ⊗ F-measurable and γ(t, T, x, ω) : [0,T]2×U×ΩisB3⊗ F-measurable;
2. EhRT 0
α(t, T) dti
<∞;
3. EhRT
0 σ(t, T)2dti
<∞;
4. Eh RT
0
R
Rγ(t, T, x)2ν(dx)dti
<∞.
ThenU =U([0,T]3×[0,T]3×U)is the class of triplets(α(t, T), σ(t, T), γ(t, T, x))∈ Lucp∩U∗([0,T]3×[0,T]3×U). We writeU([0,T]3×[0,T]3×U) :=U([0,T]32×U) for simplicity. Also, ifα(t, T)orσ(t, T)or both are zero in a triplet we will omit writing them. That is, we write(f(t, T), γ(t, T, x))∈ U([0,T]32×U)if one of the functions is zero, and simplyγ(t, T, x)∈ U([0,T]32×U)if both are zero. If the triplet is not given by a parameterT, then the space is reduced toU([0,T]3×U).
Remark ([ØS07]).DefineM(t) := Rt 0
R
Rγ(s, T, x) ˜N(dx, ds), and let T ≤ T. Then
• M(t)is a local martingale ont≤ T if Z t
0
Z
R
γ2(s, T, x)ν(dx)ds <∞;
• M(t)is a martingale ont≤ T if
E
"
Z t 0
Z
R
γ2(s, T, x)ν(dx)ds
#
<∞.
Itô formula for Itô-Lèvy processes is an inevitable theorem when we want to interchange between a Itô-Lévy process and its dynamics. We will use this theorem several times throughout the thesis.
Theorem 2.3.8 (The one-dimensional Itô formula, [ØS07]).Suppose that X(t)is an Itô-Lévy process as defined in Eq. (2.6). Letf ∈C2(R2)and define
Y(t) =f(t, X(t)). ThenY(t)is again an Itô-Levy process and dY(t) = ∂f
∂t(t, X(t))dt+∂f
∂x(t, X(t)) α(t)dt+β(t)dW(t) +1
2β2(t)∂2f
∂x2(t, X(t))dt +
Z
|x|<R
f(t, X(t−) +γ(t, x))−f(t, X(t−))
−∂f
∂x(t, X(t−))γ(t, x)
ν(dx)dt
+ Z
R
f(t, X(t−) +γ(t, x))−f(t, X(t−))N¯(dt, dx).
Remark.IfR= 0thenN¯ =Neverywhere, and ifR=∞thenN¯ = ˜N, as long as the sufficient conditions are satisfied.
In the section where we introduced Brownian motions, we mentioned that geometric Brownian motions are special cases of geometric Itô-Lévy processes. Geometric Itô-Lévy processes are SDE’s on the formdX(t) = X(t−)dL(t). By use of Itô formula (Thm. 2.3.8) it is straight forward toˆ show that the solution of the geometric Brownian motion is Et(f ◦W), where we have used Nota. 2.1.2 and 2.1.3. The solution of a geometric Itô-Lévy process is calculated by use of Itô formula (Thm. 2.3.8) in App.
A.2, and we see that the stochastic exponential contain two extra terms in that case, compared to the geometric Brownian motion case. To ease the notation when we work with geometric Itô-Lévy processes, we present another notation which will give a neat representation of the stochastic exponential in that case as well.
Notation 2.3.2 (Stochastic integral II).We define the dynamics (f◦N¯) :=
Z
|x|<R
log 1 +f(t, x)
−f(t, x)
ν(dx)dt
+ Z
R
log 1 +f(t, x)N¯(dx, dt),
where N¯(t, U) is the Poisson random measure, ν is the Lévy measure, f(t, x)≥ −1andf(t, x),log 1 +f(t, x)
∈ U([0,T]3×U).
Then, by App. A.2 and Nota. 2.1.2, 2.1.3 and 2.3.2, we see that the SDE dX(t) =X(t−)dLˆhas a solution of the formX(t) =X(0)Et
f1◦W+f2◦N¯ . Another theorem which is very important for the derivations in this thesis is Girsanov’s theorem. Girsanov’s theorem makes it possible to do measure changes, and consider stochastic processes under the given probability measure. This is a powerful tool in stochastic analysis, as some situations become considerably simplified under certain probability measures. We define predictable processes before we introduce Girsanov’s theorem and Girsanov’s theorem for Itô-Lévy processes.